* The definitions of special terms used in the Introduction are given later. The following contractions are occasionally used: pol. (polynomial), el. (element), coeff. (coefficient), hom. (homogeneous), p.p. (power product), lin. ind. (linearly independent).

† W. i (1930), 243 pp. and W. ii (1931), 216 pp. Verlag von Julius Springer, Berlin.

‡ Gesammelte math. Werke, Vieweg und Sohn, Brunswick.

§ American J. of Math. 4 (1881), 97–229.Google Scholar

∥ Proc. Lond. Math. Soc. (2), 6 (1908), 77–118.Google Scholar

¶ Math. Annalen, 60 (1905), 20–116.Google Scholar

** T. “Modular Systems”, Cambridge Tracts in Maths. and Math. Physics, No. 19 (1916), 112 pp.Google Scholar A modular system is now more precisely named a pol. ideal. It consists of the aggregate of pols. A ₁F ₁+A ₂F ₂+…+A _kF _k, where F ₁,…, F _k are given pols. in n unknowns x ₁, x ₂,…, x _n and A ₁,…, A _k are arbitrary pols. It is specialised in respect to the coeffs. which are els. of a corpus, and so link it up directly with geometry.

†† N. (i). “Idealtheorie in Ringbereichen”, Math. Annalen, 83 (1921), 25–66.Google Scholar

N. (ii). “Abstrakter Aufbau der Idealtheorie”, Math. Annalen, 96 (1926), 26–61.Google Scholar

N. (iii). “Hyperkomplexe Grössen und Darstellungstheorie.” Math. Zeitschrift, 30 (1929), 641–692;Google Scholar including idealtheory of non-commutative rings.

* The proof that the two forms of the postulate are equivalent is given in W. ii, p. 25 f. An ideal M ₂includes another M ₁ if M ₂ includes all the els. of M ₁; and M ₁ ⊂ M ₂ denotes that M ₂ includes M ₁ and other els. in addition. This is looking at M ₁ and M ₂ as aggregates. Regarding them as ideals and entities, M ₂ is called a proper factor of M ₁, and M ₁ a proper multiple of M ₂.

† Example. In the ring of the natural numbers, and in the case of pols, in one unknown, every ideal has a basis (φ) consisting of a single element. Hence M ₁ ≡ (φ), and between (0) and (φ) we can insert any number of ideals

But only a finite number of ideals can be inserted between (φ) and R, viz. if φ is the product of k irreducible factors not more than k − 1 ideals can be inserted in the chain between (φ) and R.

* This is more than sufficient for a definition. No attempt is made here or later to analyse a definition.

† We say that the els. of an aggregate admit of multiplying or adding when any product or sum of its els. is also an el. If ab = c then a = cb ⁻¹ and b = a ⁻¹c.

* Noether's definition (N. (i), p. 25), avoiding any reference to a prime ideal. Lasker's definition for pol. ideals was dependent on the manifold of Q and the prime P determined by it.

* Assuming that no pol. in the ring vanishes unless all its coeffs. vanish.

* The proofs by Lasker and in T. § 56 are both unsatisfactory.

* In T. square brackets are used for the basis of M ⁻¹ but here round brackets are used, just as for M. While (E ₁,…,E _k) denotes the sum or h.c.f. of (E ₁),…, (E _k), (E ₁, …, E _k)⁻¹ is the l.c.m. of (E ₁)⁻¹, …, (E _k)⁻¹, viz.

M ≡ (E ₁, …, E _k)⁻¹≡[(E ₁)⁻¹, …, (E _k)⁻¹].

* Also the els. of (M, u ₀, u ₁) are subject to five or four conditions, according as u ₁ passes through the vertex of M or not; i.e. the apparent extra point of (M, u ₀) is retained, or not, in (M, u ₀, u ₁) in the two cases.

* Also a non-H-ideal is superperfect if its equivalent H-ideal is superperfect. An H-ideal of rank n which is a principal system is superperfect. is not; for and

† A whole el. of M ^(r) or (M ^(r))⁻¹ is, as the name indicates, an el. whose coeffs. are whole instead of fractional as regards x _r+1, …, x _n. A whole el. or basis of M ^(r) is simply an el. or basis of M when M is unmixed. A whole basis of (M ^(r))⁻¹ is a basis of whole els. such that any whole el. of (M ^(r))⁻¹ is of the form where A ₁, …, A _k are pols. in K[x ₁, …, x _n].

* Hence the least number k of els. which will suffice for a basis of M″ ≡ M: M′ (over and above els. of M) is independent of the choice of M, the ideal M′ being given.